#394296

order	512
nilpotency class	3
isoclinism	isoclinic class of #394108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14716
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15)(4,21)(5,26)(6,14)(7,12)(8,13)(9,18)(10,24)(11,27)(16,34)(17,46)(19,33)(20,32)(22,35)(23,37)(25,36)(28,39)(29,38)(30,45)(31,49)(40,43)(41,47)(42,61)(44,56)(48,57)(50,51)(52,54)(53,59)(55,64)(58,62)(60,63);(1,3,6,18,8,20,7,19)(2,9,12,15,13,33,14,32)(4,16,23,43,25,45,24,44)(5,17,27,47,28,49,29,48)(10,34,36,56,37,30,21,40)(11,31,26,41,38,46,39,57)(22,42,51,62,53,63,52,64)(35,58,54,61,59,55,50,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30)(11,35)(12,36)(13,37)(14,21)(15,40)(17,42)(18,43)(19,44)(20,45)(26,54)(27,51)(28,53)(29,52)(31,55)(32,56)(33,34)(38,59)(39,50)(41,60)(46,58)(47,62)(48,64)(49,63)(57,61);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,31)(10,35)(12,26)(13,38)(14,39)(15,41)(16,42)(18,47)(19,48)(20,49)(21,50)(23,51)(24,52)(25,53)(30,55)(32,57)(33,46)(34,58)(36,54)(37,59)(40,60)(43,62)(44,64)(45,63)(56,61)

Polycyclic presentation:

2^9|b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285493, #72285536, C₂×#394296, …
maximal normal subgroup of	#72285493, #72285536, C₂×#394296, …
maximal quotient of	C₂×#394296

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301492);